TD3

o3
m, x, F =−kx.
2
E x p.
x(t), p(t)
ω
h
n, E.
En.
Eexact
n=~ω(n+ 1/2) n= 0,1,2. . .
EnEexact
n
ρ(E) = dn/dE ρ(E)
L
V(x)?
Eexact
n= (n2π2/L2)(~2/2m)n= 1,2, . . .
V
ρ(E) = dn
dE
N
N m
V. N '1023

N
E
V ol (E)
E
n(E)
E
Nln n'ln ρ'ln dn
S(E) = kln ρ(E)S N
rRn
x2
1+x2 2.. +x2
n=r2
Vn(r) = αnrn
αn=πn/2
Γ(n
2+ 1) ≈2πe
nn/21
√πn n1
Γ() Γ(n) = (n−1)!;
Γ(1/2) = √π; Γ(x+1) = xΓ(x).

N
6N
x1, px1, y1, py1, z1, pz1
| {z }
1
, . . . , xN, pxN, yN, pyN, zN, pzN
| {z }
N
6N
E=PN
i=1
~p2
i
2m=1 2mPN
i=1 p2
xi+p2 yi+p2 zi
N
X
i=1 p2
xi+p2 yi+p2 zi≤2mE
3N
r2= 2mE
VSphere =α3Nr3N/2
V N
VN
V ol (E) = α3Nr3N/2VN=α3N(2mE)3N/2VN
N
∆x1∆px1
| {z }
h
,...,∆zN∆pzN
| {z }
h
=h3N
n(E) = V ol(E)
h3N=h−3Nα3N(2mE)3N/2VN
' V
h32πe2mE 3N3/2!N1
√π3N
ρ(E) = dn
dE =n3N 21 E

ln ρ= ln n+ ln 3N
2−ln E
ln n∝Nln N, N '1023
ln ρ'ln n
ln dn = ln ρ−ln dE 'ln ρ'ln n
S(E) = kln n=kN ln V
h32πe2m 3
E
N3/2!